Compactified Jacobians of extended ADE curves and Lagrangian fibrations

Abstract

We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalize Kodaira’s classification of singular elliptic fibers and thus call them extended ADE curves. On such a curve [Formula: see text], we describe a compactified Jacobian and show that its components reflect the intersection graph of [Formula: see text]. This extends known results when [Formula: see text] is reduced, but new difficulties arise when [Formula: see text] is non-reduced. As an application, we get an explicit description of general singular fibers of certain Lagrangian fibrations of Beauville–Mukai type.